1. Exact Analytical Formulation for coordinated motions in Polypeptide Chains Vageli Coutsias*, Chaok Seok** and Ken Dill** with applications to: Fast Exact Loop Closure in Homology Modeling Monte Carlo Minimization for Conformational Search Small Peptide Ring Efficient Conformational Search * Department of Mathematics and Statistics, University of New Mexico ** Department of Pharmaceutical Chemistry, UCSF

2. Study of localized motions in a polypeptide chain

3. C1C1 C2C2 C3C3 N3 N2 N1 C’1 C’2 C’3 δ LOOP CLOSURE: find all configurations with two end-bonds fixed The angle between the planes N1-C  1-C  3 and C  1-C  3-C’3 is given, the orientation of the two fixed bonds (N1-C  1 and C  3-C’3) wrt the plane C  1-C  2-C  3 can assume several values (at most 8 solutions are possible)

4. Peptide: the elemental unit

5. 1.32 114 123 1.47 1.53 3.80 122 119 1.24 1. A Canonical Peptide unit (trans configuration) in the body frame (after Flory)

7. The pep-2 “capstone”

8. With the base and the lengths of the two peptide virtual bonds fixed, the vertex is constrained to lie on a circle. Tripeptide Loop Closure Bond vectors fixed in space Fixed distance

9. In the body frame of the three carbons, the anchor bonds lie in cones about the fixed base. Given the distance and angle constraints, three types of virtual motions are encountered in the body frame

10. C1C1 C2C2 C3C3 N3 N2 N1 C’1 C’2 C’3 δ Transferred motions in the body frame of three contiguous C  carbon units: In this frame the C  carbons resemble spherical 4-bar linkage joints

11. Motion type 1: peptide axis rotation With the two end carbons fixed in space, the peptide unit can rotate about the virtual bond

12. Motion type 2: Coordinated rotation at junction of 2 rotatable bonds (the angle between the two bonds remains fixed as each rotates about its own peptide virtual axis). 4 3 2 1

13. Crank Follower Two-revolute, two-spheric-pair mechanism

14. The general RSSR linkage

15. The 4-bar spherical linkage

16. z d x y

17. z d x y -3.5-3-2.5-2-1.5-0.500.5 0 1 1.5 2 2.5 3 are constrained to lie on the circles fixed The use of intrinsic coordinates distinguishes our method from other exact loop closure methods (Wedemeyer & Scheraga ‘00, Dinner ‘01)

18. Brickard (1897): convert to polynomial form via

19. y z x R1 R2 4 3 2 1 A complete cycle through the allowed values for  (dihedral (R1,R2) -(L1,R1) )and  (dihedral (R1,R2)-(L2,R2)) L1 L2

22. Differential equations for the reciprocal angles,  and . Fixed angle between the two bonds, C  N and C  C’:

23.  =.81=  2  t

24.  =.81=  2   =.4,  2=.81

25. A stressed peptide in the body frame of the virtual bonds P(n-1)—P(n) Motion type 1: Peptide axis Rotation (rigid)

26. Motion type 2: Coordinated rotation at junction of 2 rotatable bonds

27. Definitions

28. Solution Closure requires: Label: Branch present if

30. 8 real solutions at most Numerical evidence only

33. The transformation 3 coupled polynomials: Common (real) zeros give feasible solutions.

34. Method of resultants gives an equivalent 16 th degree polynomial for a single variable Numerical evidence that at most 8 real solutions exist. Must be related to parameter values: the similar problem of the 6R linkage in a multijointed robot arm is known to possess 16 solutions for certain ranges of parameter values (Wampler and Morgan ’87; Lee and Liang ‘’89).

35. Methods of determining all zeros: (1) carry out resultant elimination twice; derive univariate polynomial of degree 16 solve using Sturm chains and deflation (2) carry out resultant elimination once convert matrix polynomial to a generalized eigenproblem of size 24 (3) work directly with trigonometric version; use geometry to define feasible intervals and exhaustively search. It is important to allow flexibility in some degrees of freedom

36. input coeff sturm coord tot 8 0.067 0.253 3.814 0.442 4.576 0 0.084 0.253 0.141 0.478 4 0.085 0.252 6.513 0.218 7.068 4 0.088 0.252 3.392 0.228 3.960 0 0.066 0.296 0.138 0.500 2 0.066 0.293 0.356 0.115 0.830 2 0.085 0.253 0.411 0.124 0.873 4 0.067 0.253 1.957 0.227 2.504 4 0.067 0.252 0.582 0.219 1.120 2 0.067 0.251 1.803 0.114 2.235 2 0.066 0.253 0.438 0.141 0.898 6 0.067 0.257 2.041 0.321 2.686 6 0.066 0.254 2.131 0.322 2.773 2 0.066 0.251 0.336 0.115 0.768 2 0.067 0.252 1.726 0.115 2.160 2 0.068 0.250 0.332 0.115 0.765 2 0.067 0.251 1.678 0.115 2.111 0 0.067 0.251 0.138 0.456 4 0.066 0.252 6.360 0.218 6.896 4 0.068 0.253 1.870 0.219 2.410 Timings for loop closure by reduction to 16 th degree polynomial; zero localization via Sturm’s method. Successively solve loop closure by successively removing the two peptide units adjacent to each C  carbon in a chain of known conformation. Loop closure should reproduce original, however off canonical structures do abound. Zero solutions indicate that the closure was not possible with canonically configured backbone, i.e. there was a deformation of some bond angles or  dihedrals in the original strucure.

37. input matr gen_e coord tot 8 0.067 0.069 1.096 0.431 1.663 0 0.067 0.069 3.748 3.884 4 0.067 0.068 2.036 0.221 2.392 4 0.067 0.069 2.741 0.223 3.100 0 0.067 0.069 3.735 3.871 2 0.067 0.068 3.690 0.118 3.943 2 0.067 0.069 3.711 0.119 3.966 4 0.067 0.068 3.656 0.220 4.011 4 0.066 0.069 2.032 0.225 2.392 2 0.068 0.068 3.206 0.119 3.461 2 0.065 0.069 3.710 0.118 3.962 6 0.067 0.068 1.984 0.329 2.448 6 0.066 0.069 1.710 0.326 2.171 2 0.067 0.073 3.215 0.118 3.473 2 0.069 0.068 3.700 0.117 3.954 2 0.068 0.069 2.789 0.116 3.042 2 0.067 0.069 2.785 0.119 3.040 0 0.067 0.068 3.760 3.895 4 0.067 0.068 2.373 0.219 2.727 4 0.067 0.069 2.033 0.220 2.389 Timings for loop closure via reduction to 24x24 generalized eigenproblem.

38. Application to loop sampling No. of loopsNumerical closureAnalytical closure generated (best RMSD) 151 (1.61) 40,000 (1.23) accepted (best RMSD) 1 (6.75) 1,374 (1.58) Analytical closure of the two arms of a loop in the middle Comparison: 10 residue loop sampling (Matt Jacobson)

39. 1r69.pdb

40. 1r69: Res 9-19 alternative backbone configurations

41. The 3 fixed points/3 virtual axes transform can be found among any three Ca atoms, anywhere along the chain

43. Motion type 3: Dihedral rotation (actual move, length changing; not limited to - type dihedrals) (quaternion notation)

44. Altering an internal dihedral leads to a “nearby” loop closure problem. A sequence of small changes results in a continuous family of deformations (shown here as applied to the deformation of a disulfide bridge).

48. Refinement of 8 residue loop (84-91) of turkey egg white lysozyme Native structure (red) and initial structure (blue) Baysal, C. and Meirovitch, H., J. Phys. Chem. A, 1997, 101, 2185

49. The continuous move: given a state assume D2b, D4a fixed, but D3 variable tau2  sigma4 determined by D3 (1) tau1  sigma2, tau4  sigma5 trivial (2) alpha1, alpha5 variable but depend only on vertices as do lengths (lengths 1-2, 1-5, 4-5 are fixed) Given these sigma1  tau1, sigma5  tau5 known (sigma1  tau5 given) (3) Dihedral (2-1-5-4) fixes remainder: alpha2, alpha4 determined (sigma2  tau2, sigma4  tau4 known)

50. 2 1 3 B A C Designing a 9-peptide ring pep virtual bond 3-pep bridge design triangle cysteine bridge 9-pep ring Modeling R. Larson’s 9-peptide

51. 3 peptide units are placed at the vertices of a triangle with random orientations, and they are connected by exact loop closure. The max and min values of the 3-pep bridge set the limits for the sides of the triangle. Designing a 9-peptide ring

52. In designing a 9-peptide ring, the known parameters of 2-pep bridges (and those of the S2 bridge, if present) are incorporated in the choice of the foundation triangle, with vertices A,B,C (3 DOF) B A C

53. B A C peptide virtual bond (3 dof for placement)x3=9 2-pep virtual bond (at most 8 solutions) design triangle sides (3 dof )

54. 8-2-4 4-6-2 4-2-4 4-2-2 Cyclic 9-peptide backbone design Numbers denote alternative loop closure solutions at each side of the brace triangle

55. Using backbone kinematics in combination with efficient (clever) placement of sidechains can be used in a “rational approach for exploring conformation space. The 3 fixed points/3 virtual axes transform can be used as a means of enforcing constraints (such as loop closure). It can be used to generate minimum- Distortion moves for Monte Carlo energy minimization. Generalizations where one pair is disjointed are also possible with a simple solution as well.

56. THANK YOU! Visits to UCSF where much of the work was performed supported in part by a NIH grant to Ken Dill Cyclic peptide modeling: inspired by conversations with Michael Wester, R. Larson Animations: Raemon Gurule, Carl Mittendorff, Heather Paulsen and Marshall Thompson (math. 375, Spring 02 class project) References Analytical loop closure Wedemeyer and Scheraga J Comput Chem 1999 Go and Scheraga Macromolecules 1978 Dinner J Comput Chem 2000 Bruccoleri and Karplus Macromolecules, 1985 Coutsias, Seok, Jacobson and Dill (preprint) 2003 Mechanisms Hartenberg and Denavit 1964 Hunt Oxford 1990 Duffy 1980 Numerical Methods Manocha, Appl. of Comput. Alg. Geom., AMS,1997 Wampler and Morgan Mech Mach Theory 1991 Lee and Liang Mech Mach Theory 1988

57. General Proline Glycine Ramachandran regions